AP Calculus Area

Area of a Plane Region Calculus was built around two problems –Tangent line –Area

Area To approximate area, we use rectangles More rectangles means more accuracy

Area Can over approximate with an upper sum Or under approximate with a lower sum

Area Variables include –Number of rectangles used –Endpoints used

Area Regardless of the number of rectangles or types of inputs used, the method is basically the same. Multiply width times height and add.

Upper and Lower Sums An upper sum is defined as the area of circumscribed rectangles A lower sum is defined as the area of inscribed rectangles The actual area under a curve is always between these two sums or equal to one or both of them.

Area Approximation We wish to approximate the area under a curve f from a to b.We wish to approximate the area under a curve f from a to b. We begin by subdividing the interval [a, b] into n subintervals. Each subinterval is of width.

Area Approximation

We wish to approximate the area under a curve f from a to b. We begin by subdividing the interval [a, b] into nWe begin by subdividing the interval [a, b] into n subintervals of width subintervals of width. Minimum value of f in the ith subinterval Maximum value of f in the ith subinterval

Area Approximation

So the width of each rectangle is

The height of each rectangle is either or Area Approximation

So the width of each rectangle is The height of each rectangle is either or So the upper and lower sums can be defined as Lower sum Upper sum

Area Approximation It is important to note that Neither approximation will give you the actual area Either approximation can be found to such a degree that it is accurate enough by taking the limit as n goes to infinity In other words